The Bantams entertain Aldershot in front of the BT Sport cameras on Wednesday evening for the right to host Chesham United in the second round.
Burke returned to City on Monday after a spell in France with England Under 20s but he unfortunately picked up a minor problem in training on his first day back.
The young West Ham United loanee was an unused sub for England's first game with the French before the second scheduled match was cancelled due to the tragic Paris terror attacks last Friday evening.
Lee Evans and James Meredith are currently still away on international duty so they definitely won't feature against Aldershot on Wednesday.
Parkinson said: "Reece has actually picked up a slight injury in training yesterday (Monday).
"Unfortunately that makes him a doubt for Wednesday's game.
"It is a shame if he misses out because he's gone away with England Under 20s and not got any game-time so he is itching to play.
"Equally though, Nathan Clarke was excellent at the weekend and ready to play again."
Along with Burke, City will check on the progress of Josh Morris ahead of Wednesday night to see if the midfielder has a chance of featuring in the tie.
After two months sidelined with a troublesome knee injury, Morris could be included among City's substitutes for the replay.
Parkinson continued: "He has trained with the squad for the past two days so we'll just make a decision on whether to include him or not in Wednesday's match.
"I'll sit down with Steve (Parkin), Matt Barrass (Head Physio) and Josh himself just to see how he feels.
"Everyone else who was available at the weekend is available again for this tie."